# Time and Frequency Domain Equalization

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1 Time and Frequency Domain Equalization Presented By: Khaled Shawky Hassan Under Supervision of: Prof. Werner Henkel

2 Introduction to Equalization Non-ideal analog-media such as telephone cables and radio channels typically distort the transmitted signal. Due to the non-ideal situation and high data rate, Inter Symbol Interference (ISI) is introduced to the received signal. ISI arises in all pulse-modulation systems, including Pulse Amplitude Modulation (PAM) frequency-shift keying (FSK), phase-shift keying (PSK) and quadrature amplitude modulation (QAM).

3 Data Transmission System Transmission system: Input Bit Stream Encoder & Filter Modulator Telephone Circuit or Channel De- Modulator Filter & Equalizer T Decision Device Decoder Output Bit Stream Baseband system model: Channel impulse response:

4 The received signal is the superposition of the channel impulse response to each transmitted symbol and AWGN The received signal: rt () = xht j ( - jt) + n() t j If we sample the received signal at (kt+t o ), where t o accounts for the channel delay and sampler phase, we obtain: rt ( + kt) = xht ( 0) + Int xht ( + kt- jt) + n( t + kt) 0 k j 0 0 j k Conclusion: Interference term is proportional to a sample of the channel impulse response, h(t 0 -jt). i.e.: ISI = 0, iff h(t 0 - jt) = 0 (Zero crossing at the T - spaced interval)

5 Nyquist s first criterion When the impulse response has such uniformly-spaced zero crossings, it is said to satisfy Nyquist s first criterion. In frequency domain terms, this condition is equivalent to: H'( f) = constant for f 1/ 2T and H '( f) = H( f) + H( f 1/ T), 0 f 1/ T In practice, the effect of IS1 can be seen from a trace of the received signal on an oscilloscope with its time base synchronized to the symbol rate.

6

7 Time Domain Equalization In time domain equalization schemes a (digital) filter (FIR) is inserted in the signal path after the channel in the receiver part, to compact ISI. Types of time domain equalizers:

8 Linear Equalizers Example Transversal Zero-forcing Equalizer: (neglects the effect of noise) d N 2 ^ * = k estimated symbols C y n k n n= N where : C( z) = 1 1 F( z), from Peak Distortion Finite-length ZF equalizer is guaranteed to minimize the peak distortion or worst-case IS1 only if the peak distortion before equalization is less than 100 percent; i.e., if a binary eye is initially open or Minimum phase channel The least mean-square (LMS) equalizer is more robust, as the equalizer coefficients are chosen to minimize the MSE

9 Linear Adaptive Equalizers (LMS/RLS) LMS Coefficient Update: * C = k 1 C + + k ekxk, k = 0,1, L RLS Cost function k k n ξ = w I C X n= 0 ' * 2 n k n where weighting factor w is selected 0<w<1 Performance Comparison

10 Non-linear Equalizers Example Decision Feedback equalizer: Is useful for channels with severe amplitude distortion, uses decision feedback to cancel the interference from symbols which have already been detected.

11 Low Complexity Equalization by Guard Insertion DMT consists of a large number of QAM modulated carriers that are orthogonal. Obviously, independent transmission channels (assumed N parallel channels) are obtained only if appropriate symbol synchronization is performed at the receiver to compensate for the delay introduced by the channel Mathematical description of DMT modulation, transmission, and demodulation:

12 The Inter-carrier and Inter-symbols Interference When the cyclic prefix is shorter than the memory of the channel, inter-carrier interference (ICI) as well as ISI are unavoidable. Interference can be written as a weighted sum of the QAM symbols transmitted during the previous and next DMT symbol. In addition, because of the loss of orthogonality between the carriers in the current symbol, extra ICI is generated. The goal: Searching for low-complexity equalization schemes to obtain insight in the different interference mechanisms.

13 Interference Type in DMT Signal ICI 1 is the interference from QAM symbols transmitted over the other carriers of the m th symbol. ISI is the interference from a k m 1 and a k m+1, the QAM symbols modulating the k th carrier during the previous and next DMT symbols. ICI 2 is the interference caused by the QAM symbols in the previous and next DMT symbols, transmitted over carriers other than the k th carrier.

14 Inter-carrier Interference Two segments of the impulse response contribute to the interference: the head and tail. ICI can be expressed as a function of the FFT of the head and tail. ICI 2 is calculated from carrier i and j, and weighted by W(k-j). ICI is reciprocal, i.e.: the interference power from carrier k on j equals the interference power from carrier j on k.

15 Time Domain Equalization (TEQ) Y = H. X + n A (digital) filter is inserted in the signal path before the FFT function. Time domain equalizer (TEQ) coefficient initialization for true capacity optimization leads to a highly nonlinear optimization problem. The performance of the TEQ is not predictable, as there is no direct relation between this time domain mean-square-error (MSE)-optimal channel shortening and the transmission capacity

16 TEQ Optimization Problem The cascading of the TEQ and channel impulse response CIR ({hk}) approximately forms an FIR target impulse response (TIR) ({bk}), with an impulse response length shorter than the cyclic prefix. The relative delay between the equalizer and the TIR is denoted by The unknown parameters, {wk}, and {bk} are computed based on a mean squared error (MSE) criterion (Channel Shortening Problem), and can be found from the next figure.

17 Frequency (Per-tone) Domain Equalization In the per-tone equalization, the MSE optimization is performed for each carrier separately, leading to improved as well as more predictable and reproducible performance. Every single tone is given its own distinct and optimal equalizer. The computational requirement is roughly kept at the same level, or even smaller, as the computational requirement in a TEQ-based modem. An appropriate initialization scheme (based on equalizer training) is included. The memory requirement obviously increases significantly, but this is not considered a major implementation.

18 Frequency (Per-tone) Domain Equalization Our approach is based on transferring the TEQoperations to the frequency domain (i.e., after the FFTdemodulation). A (complex) tap FEQ for tone. Then allow each tone to have its own tap FEQ for tone i. instead of one FFT-operation per symbol, we now apparently need FFT-operations (one FFT for each column of )!!!! (Dangerous)

19 Frequency (Per-tone) Domain Equalization But: Fortunately, because of the Toeplitz structure of Y, these FFTs can be calculated efficiently by means of a sliding FFT. Only one full FFT has to be calculated (for the first column of ) and the remaining FFTs can be deduced as follows:

20 Complexity of FEQ vs TEQ We can conclude that the resulting complexities are comparable and both of order F s T. The complexity of the per tone method can be reduced even further by varying the equalizer length per tone and setting the length to zero for non-used tones.

21 Equalization Initialization For each of the used tones, we find the MMSE-FEQ (for a particular choice of ) by minimizing the following cost function:

22 Equalization Initialization For each of the used tones, we find the MMSE-FEQ (for a particular choice of ) by minimizing the following cost function: NOTE: The modified T-tap FEQ Per-tone reduces the complexity by incorporating the liner combination of the FEQ coefficients, such that the global FEQ for each tone i has as its inputs the i th output of the FFT and T-1 (real) difference terms.

23 Simulation Results ADSL simulation results are presented for downstream to compare the different equalizers. Standard channel T #9 with NEXT from 12 DSL and 12 HDSL disturbances is considered. To compute capacity, we take the SNR gap Γ=9.8 db, noise margin γ m = 6 db, coding gain γ c = 3 db. The number of bits assigned to tone i is:

24 Per-tone Equalization Application MULTIPLE-INPUT/MULTIPLE-OUTPUT EQUALIZATION Other multicarrier systems must solve equalization differently. Receiver structures have been investigated that are valid not only for IFFT-based DMT transmitters without cyclic extension, but also for any other filterbankbased multicarrier system. Is a cyclic extension is still useful? when the cyclic prefix length is shorter than the MIMO order, then cyclic prefix has no use anymore. What receiver structure could be derived if no cyclic extension was used at all (as in case of the high Order MIMO)? Swapping the filtering operations of the MIMO channel and the FFT, we can show that shown that each tone of a MIMO OFDM system can be viewed as a MIMO SC system. As a result, the existing equalization approach for MIMO SC systems can be applied to each tone of a MIMO OFDM system. This socalled Per tone TEQ (PTEQ) approach for MIMO OFDM systems.

25 Per Tone Equalization for OFDM over wireless doubly selective channel 1-Tap TEQ, to convert doubly selective channel into pure frequency selective channel. 1-Tap FEQ, that is optimized for each tone, and the performance is maximized by summing up the N tones.

26 Per Tone Equalization for OFDM over wireless doubly selective channel Important Notes: The proposed ICI mitigation technique is simply achieved by taking P FFTs of different modulated versions y^p(n) of the received sequence y(n). Note that this actually corresponds to over-sampling the received sequence in the frequency-domain by a factor of P. To detect a symbol on the k th subcarrier of the i th OFDM block, neighboring subcarriers are combined at the output of the pth FFT. The resulting outputs are subsequently combined to obtain the symbol transmitted on that subcarrier For each subcarrier, we can find the MMSE equalizer coefficients by minimizing the cost function.

27 Conclusion For the same equalizer order, the PTEQ approach always has a better performance than the TEQ approach. Moreover, it can be shown that the equalization complexity for both approaches is comparable. The performance of the PTEQ approach is a much smoother function of the synchronization delay than the performance of the TEQ approach. Hence, for the PTEQ approach the synchronization delay setting is less critical than for the TEQ approach. Since a per-tone equalizer works on the symbol level, whereas a time-domain equalizer works on the sample level, a per-tone equalizer can much more easily be designed in practice than a time-domain equalizer.

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